Isosceles Triangle
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Definition of Isosceles Triangle: An isosceles triangle is a triangle having at least two sides of equal length. The term isosceles is derived from the Greek words isos meaning equal and skelos meaning leg, emphasizing that two of the triangle's sides are of the same length, while the third side may be of a different length. The angles opposite the equal sides are also equal.
In the figure above, the sides XY and XZ are equal and consequently, angles ∠Y and ∠Z have equal measure in △XYZ, hence it is an isosceles triangle.A common real-world example of an isosceles triangle is the shape of many rooftops, where two sides of the roof are equal in length, forming the "legs" of the triangle, and the peak of the roof forms the vertex.
Properties of an Isosceles Triangle
Below are some characteristics and properties of isosceles triangles:Legs
: The defining characteristic of an isosceles triangle is that it has two sides of equal length. These sides are often referred to as the legs of the triangle.Base
: The third side of the triangle, which may not equal in length to the other two, is called the base of the triangle.Base Angles
: The angles opposite the two equal sides are also equal in measure. These angles are referred to as the base angles.Base Angle Theorem
: If two angles of a triangle are equal in measure, then the sides opposite those angles are also equal. In an isosceles triangle, this means that the base sides are equal in length.Height or Altitude
: The height or altitude of an isosceles triangle is the perpendicular line drawn from the vertex (the top point where two equal sides intersect) to the base. It bisects the base and creates two congruent right triangles.
Isosceles Triangle Formulas
Perimeter of Isosceles Triangle
To calculate the perimeter of a triangle, you need to add the lengths of all three sides together. In an isosceles triangle, there are two sides (the legs) that are of equal length () and one side (the base) that may be of a different length (). Simply add twice the length of one of the equal sides () to the length of the base () to find the perimeter of the isosceles triangle.
Therefore, the formula to calculate the perimeter is expressed as:Area of Isosceles Triangle
The formula to calculate the area of an isosceles triangle is expressed as:where is the base and is the height of the triangle.
Isosceles triangles are important in geometry and have applications in various fields, such as architecture, engineering, and trigonometry. Understanding their properties and relationships can help solve problems involving these triangles and provide insights into broader mathematical concepts.